Analysis of free boundary problems in viscous fluid mechanics: traveling waves and contact lines

About This Grant

Moving interfaces in multi-phase fluid flow occur in wide range of natural phenomena at scales from microscopic to cosmic: blood flow in elastic arteries, the surface of a cup of coffee, the surface of a body of water, solar plasma meeting the vacuum of space. They also play a key role in industrial and technological applications, from bubble formation in industrial emulsion manufacturing to Rayleigh-Taylor instabilities in fusion reactors. Mathematicians are uniquely equipped to make fundamental contributions to the understanding of these diverse and important phenomena through the analysis of the nonlinear systems of partial differential equations underlying these models. This project aims at studying fundamental questions of well-posedness and stability for of two long-standing areas of interest in interfacial mechanics: the contact line problem, and the traveling wave problem. The project also contains a plan for integrating research and education through the continued funding of an undergraduate Summer Analysis Program. While the existence of traveling wave solutions to the free boundary, incompressible Euler equations has been known for nearly a century, progress on the corresponding Navier-Stokes problem only began recently with the NSF-funded work of the principal investigator (PI) and collaborators. The project aims to continue this work and demonstrate the robustness of traveling wave phenomena through the construction of solutions in more general contexts and a deeper study of the solutions. It is important to account for viscosity because, while many fluids have small viscosity (or more precisely, the fluid configuration has large Reynolds number), small is not zero, so all fluids experience some viscous effects. Developing the viscous theory also opens the possibility of connecting the viscous and inviscid cases through vanishing viscosity limits, which could potentially yield insight into the plethora of known inviscid solutions. In particular, it could lead to a selection mechanism for physically relevant inviscid solutions. The contact line problem is one of the oldest and most challenging open problems in interfacial mechanics. The analysis detailed in this project, which builds on prior NSF-funded work of the PI, will verify the PDE soundness of a recently proposed continuum model. This has the potential to make a serious impact in the science of triple-phase junctions, to open many new lines of research, and to further the applications of contact line dynamics. From an analytic perspective, it will also create and further develop bridges between energy methods, the functional calculus of differential operators, and elliptic regularity theory, which will be useful in many other contexts. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.